Question

given: ∠bcd is a right angle; ∠acb ≅ ∠cad; a is in the interior of ∠bcd
prove: ∠cad is complementary to ∠acd

statements

1. ∠bcd is a right angle.
2. ∠acb ≅ ∠cad
3. a is in the interior of ∠bcd.
4. ( mangle bcd = 90^circ )
5. ( mangle acb + mangle acd = mangle bcd )
6. ( mangle acb + mangle acd = 90^circ )
7. ( angle acb ) is complementary to ( angle acd )
8. ( angle cad ) is complementary to ( angle acd )

reasons
1.
2.
3.
4.
5.
6.
7.
8.

Explanation:

Step1: Identify given info

1. Given (as stated in problem)

Step2: Identify given congruence

1. Given (as stated in problem)

Step3: Identify interior info

1. Given (as stated in problem)

Step4: Define right angle measure

1. Definition of a right angle (a right angle has measure \( 90^\circ \))

Step5: Angle addition postulate

1. Angle Addition Postulate (if a point is in the interior of an angle, the sum of the two smaller angles equals the larger angle)

Step6: Substitute \( m\angle BCD \)

1. Substitution Property (replace \( m\angle BCD \) with \( 90^\circ \) from step 4)

Step7: Define complementary angles

1. Definition of complementary angles (two angles are complementary if their measures sum to \( 90^\circ \))

Step8: Substitute congruent angles

1. Substitution Property (replace \( \angle ACB \) with \( \angle CAD \) since they are congruent, so their measures are equal)

Answer:

1. Given
2. Given
3. Given
4. Definition of a right angle
5. Angle Addition Postulate
6. Substitution Property (using step 4 in step 5)
7. Definition of complementary angles
8. Substitution Property (using step 2 in step 7)